You need to buy some filing cabinets. You know that Cabinet X costs $10 per
unit, and requires six square feet of floor space. Cabinet Y costs $20 per unit,
and requires eight square feet of floor space. You have been given $140 for this
purchase, though you don't have to spend that much. The office has room for no
more than 72 square feet of cabinets. Let x = number of model X cabinets
purchased and y= number of model Y cabinets purchased
A. If Cabinet X holds 8 cubic feet of files, and Cabinet Y holds 12 cubic feet
of files, how many of which model should you buy, in order to maximize
storage volume?
B. Find inequalities also
A. Let's start by finding the objective function, which represents the storage volume.
Objective function: Volume = 8x + 12y
Now let's deal with the constraints.
1. Cost constraint: The total cost should be less than or equal to $140.
10x + 20y ≤ 140
2. Floor space constraint: The total floor space should be less than or equal to 72 square feet.
6x + 8y ≤ 72
3. Non-negativity constraint: The number of cabinets cannot be negative.
x, y ≥ 0 (x and y are non-negative)
Now we have the following system of inequalities:
10x + 20y ≤ 140
6x + 8y ≤ 72
x, y ≥ 0
B. The inequalities are as follows:
10x + 20y ≤ 140 (Cost constraint)
6x + 8y ≤ 72 (Floor space constraint)
x, y ≥ 0 (Non-negativity constraint)
These inequalities represent the conditions that need to be satisfied to find the optimal solution for the filing cabinet purchase.
A. To maximize storage volume, we need to optimize the cubic feet of files capacity. Let's calculate the cubic feet capacity of each cabinet first.
Cabinet X: 8 cubic feet
Cabinet Y: 12 cubic feet
We want to maximize the total storage volume, given the constraints of the budget and floor space. Let's set up the objective function and constraints.
Objective function:
Maximize V = 8x + 12y (Storage volume in cubic feet)
Constraints:
10x + 20y ≤ 140 (Budget constraint)
6x + 8y ≤ 72 (Floor space constraint)
Now let's graph the feasible region and identify the optimal point.
B. Inequalities:
Budget constraint: 10x + 20y ≤ 140
Floor space constraint: 6x + 8y ≤ 72
These inequalities represent the constraints on the budget and floor space. So the number of each model should satisfy these inequalities.
A. To maximize the storage volume, we need to determine the number of each model to buy. Let's assume we buy x units of Cabinet X and y units of Cabinet Y.
The storage volume of Cabinet X is 8 cubic feet per unit, so the total storage volume of x units of Cabinet X would be 8x cubic feet.
Similarly, the storage volume of Cabinet Y is 12 cubic feet per unit, so the total storage volume of y units of Cabinet Y would be 12y cubic feet.
We want to maximize the storage volume, so we can formulate our objective function as:
f(x,y) = 8x + 12y
Now, we need to consider the given constraints:
1. Cost constraint: We have $140 available for the purchase. The cost of Cabinet X is $10 per unit, and the cost of Cabinet Y is $20 per unit. So, the cost constraint can be written as:
10x + 20y ≤ 140
2. Space constraint: The office has room for no more than 72 square feet of cabinets. The floor space required by Cabinet X is 6 square feet per unit, and the floor space required by Cabinet Y is 8 square feet per unit. Therefore, the space constraint can be written as:
6x + 8y ≤ 72
To find the number of each model to maximize the storage volume, we can solve this linear programming problem by graphing the feasible region and identifying the optimal solution or by using linear programming software.
B. The inequalities for the given constraints are:
1. Cost constraint: 10x + 20y ≤ 140
2. Space constraint: 6x + 8y ≤ 72
These inequalities represent the limits on the cost and space for the purchase of filing cabinets.